Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L(w)$

Abstract

Given a Muckenhoupt weight $w$ and a second order divergence form elliptic operator $L$, we consider different versions of the weighted Hardy space $H^1_L(w)$ defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by $L$. We show that all of them are isomorphic and also that $H^1_L(w)$ admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Hofmann and S. Mayboroda in [19] and we can immediately recover the unweighted case. Some of our tools consist in establishing weighted norm inequalities for the non-tangential maximal functions, as well as comparing them with some conical square functions in weighted Lebesgue spaces.